1. Field of Disclosure
Embodiments described herein generally relate to an improved inflow performance relationship model for oil reservoirs and a computer-implemented method for estimating the inflow performance. Specifically, a generalized model and computer-implemented method is provided that predicts the performance of oil wells having a slant angle within the entire azimuth of zero degrees to ninety degrees for both saturated and under saturated reservoirs.
2. Description of the Related Art
The background description provided herein is for the purpose of generally presenting the context of the disclosure. Work of the presently named inventors, to the extent the work is described in this background section, as well as aspects of the description that may not otherwise qualify as prior art at the time of filing, are neither expressly nor impliedly admitted as prior art against the present disclosure.
Several studies have been conducted that aim to predict the performance of oil wells in order to determine an optimum production strategy to maximize recovery from the reservoirs. Due to the enormous productivity of slanted/horizontal oil wells as compared to conventional vertical wells, it is important that the performance of slanted oil wells be accurately estimated in order to aid decision making such as increasing or decreasing the production rate of the well, whether to stimulate or fracture the well and the like.
Various models have been developed to estimate productivity of vertical and horizontal wells, but the productivity of horizontal wells has been the subject of most recent investigations. Generally the inflow performance relationship (IPR) is a mathematical tool used to assess well performance by plotting the well production rate against the flowing bottom hole pressure (BHP). The data required to create the IPR are obtained by measuring the production rates under various drawdown pressures. The reservoir fluid composition and behavior of the fluid phases under flowing conditions determine the shape of the curve. A model described by Vogel, “Inflow Performance Relationships for Solution-Gas Drive Wells,” JPT (January, 1968), pp. 83-92, incorporated by reference herein, is a fundamental model for vertical wells. Vogel's IPR is formulated as:
            q      o              q              o        ,        max              =      1.0    -          0.2      ⁢              (                              P            wf            ′                                P            r                          )              -          0.8      ⁢                        (                                    P              wf              ′                                      P              r                                )                2            
Wherein qo is the oil flow rate, Pr is the average reservoir pressure and P′wf is the bottom-hole flowing pressure.
The work conducted by Standing, “Inflow Performance Relationships for Damaged Wells Producing by Solution-Gas Drive,” JPT, November 1970, pp. 1399-1400, incorporated herein by reference, modified Vogel's IPR to account for formation damage or stimulation by including the effect of skin factor through the concept of Flow Efficiency (FE). Further Standing extended the application of Vogel's IPR to predict future IPR of a well as a function of reservoir pressure. The IPR for non-zero skin factor is given as:
                    q        o                    q                  o          ,          max                          FE          =          1                      ⁢    1.8    ⁢          FE      ⁡              (                  1          -                      (                                          P                wf                                            P                r                                      )                          )              -      0.8    ⁢                            FE          2                ⁡                  (                      1            -                          (                                                P                  wf                                                  P                  r                                            )                                )                    2      
Wherein Pwf is the bottom-hole pressure with a non-zero skin factor.
Fetkovich, “The Isochronal Testing of Oil Wells,” Paper SPE 4529, presented at the 48th Annual Fall Meeting, Las Vegas, Nev., Sep. 30-Oct. 3, 1973. (SPE Reprints Series No. 14, 265)”, incorporated herein by reference, developed an IPR for under-saturated oil reservoirs using two scenarios: when the bottom-hole flowing pressure is above bubble-point pressure, a straight line IPR is used qo=J(Pr−Pwf), wherein J denotes the productivity index of the well. When the bottom-hole flowing pressure is below bubble-point pressure, the IPR is given as:
      q    o    =            J      ⁡              (                              P            r                    -                      P            b                          )              +                  1                  2          ⁢                      P            b                              ⁢              (                              P            b            2                    -                      P            wf            2                          )            
Cheng, in “Inflow Performance Relationships for Solution-Gas-Drive Slanted/Horizontal Wells,” Paper SPE 20720 presented at the 1990 SPE Annual Technical Conference and Exhibition, New Orleans, La., September 23-26”, incorporated herein by reference, presented IPR equations for slanted and horizontal wells. The IPR for all the slant angles considered by Cheng can be generally expressed as:
            q      o              q              o        ,        max              =            a      ⁢                          ⁢      0        -          a      ⁢                          ⁢      1      ⁢              (                              P            wf            ′                                P            r                          )              -          a      ⁢                          ⁢      2      ⁢                        (                                    P              wf              ′                                      P              r                                )                2            
Wherein qo,max is the maximum oil flow rate through the well. The constants a0, a1 and a2 vary for different slant angles. The following assumptions were made by Cheng in his investigation: (1) the well is located in the center of a completely bounded reservoir of rectangular prism geometry, (2) the reservoir is homogeneous and isotropic with constant water saturation, (3) water saturation is immobile during depletion of the well. Therefore, only two phase flows (oil and gas) are considered in the reservoir, (4) the well is producing under a semi-steady state condition and (5) capillary pressure forces of reservoir fluids are neglected.
Further, the work conducted by Beggs “Production Optimization Using NODAL Analysis,” OGCI, Tulsa, Okla., 1991, incorporated by reference herein, modified Vogel's IPR to be suitable for under-saturated oil reservoirs by using rate tests for two different cases. The rate test for the first case was performed when the bottom-hole flowing pressure is equal to or greater than the bubble-point pressure, while the rate test for case two was performed when the bottom-hole flowing pressure is less than the bubble-point pressure. Specifically, for Case I: Pb<Pwf<Pr, the following equation is stated for the flow rateqo=J(Pr−P′wf), when Pwf>Pb 
For Case II: Pwf<Pb, the flow rate is expressed as
      q    o    =            J      ⁡              (                              P            r                    -                      P            b                          )              +                            JP          b                1.8            ⁡              [                  1          -                      0.2            ⁢                          (                                                P                  wf                  ′                                                  P                  b                                            )                                -                      0.8            ⁢                                          (                                                      P                    wf                    ′                                                        P                    b                                                  )                            2                                      ]            
Also, when Pwf<Pb<Pr, the productivity index is expressed as
  J  =            q      o                      P        r            -              P        b            +                                    JP            b                    1.8                ⁡                  [                      1            -                          0.2              ⁢                              (                                                      P                    wf                    ′                                                        P                    b                                                  )                                      -                          0.8              ⁢                                                (                                                            P                      wf                      ′                                                              P                      b                                                        )                                2                                              ]                    
Wherein for all Pwf<Pb, the flow rate can be expressed as
      q    o    =            J      ⁡              (                              P            r                    -                      P            b                          )              +                            JP          b                1.8            ⁡              [                  1          -                      0.2            ⁢                          (                                                P                  wf                  ′                                                  P                  b                                            )                                -                      0.8            ⁢                                          (                                                      P                    wf                    ′                                                        P                    b                                                  )                            2                                      ]            
For Pwf>Pb the flow rate is expressed as qo=J(Pr−P′wf). Further, to account for formation damage or stimulation, Beggs' IPR was modified to produce a flow rate as stated below:
      q    o    =            J      ⁡              (                              P            r                    -                      P            b                          )              +                            JP          b                1.8            ⁡              [                              1.8            ⁢                          (                              1                -                                  (                                                            P                      wf                                                              P                      b                                                        )                                            )                                -                      0.8            ⁢                                          FE                ⁡                                  (                                      1                    -                                          (                                                                        P                          wf                          ′                                                                          P                          b                                                                    )                                                        )                                            2                                      ]            
The above models characterize flow rate for vertical wells in under-saturated reservoirs. Specifically, the flow rate computations are determined in wells wherein the average reservoir pressure is below a bubble point pressure. Accordingly there is a requirement for a generalized model for determining oil well performances of under saturated and saturated reservoirs.